In a thermomechanical analysis, frequently also designated as TMA (“thermomechanical analysis”), one or more mechanical properties of a sample of a material are measured as a function of the temperature.
The material can be, for example, a solid, liquid or pasty material. Frequently here, for example, a length variation is measured as a function of the temperature and/or time under well-defined mechanical loading of the sample, wherein the sample is usually exposed to a predetermined time-dependent temperature, i.e. a “temperature programme”.
In particular, in the case of an almost negligible (relatively small) mechanical loading of the sample, e.g. only caused by using a sensing stamp or the like to measure the length variation, the TMA is frequently also designated as dilatometry or when using a modulated temperature program as temperature-modulated dilatometry.
In a TMA of the type of interest here, a “modulated temperature programme” is used, which means that the time profile of the temperature is composed of a basic, usually linearly predefined temperature variation corresponding to a “basic heating rate” and of a usually sinusoidal (alternatively feasible e.g. triangular, rectangular or sawtooth-shaped) predefined temperature modulation superimposed on this temperature variation.
An important aim of a TMA of the type of interest here is to determine for the material of the sample at least one “reversible component” of the length variation (caused by temperature variation) and/or in particular e.g. a reversible component of the coefficient of thermal expansion.
Since, if the sample is exposed to a thermally induced conversion process (e.g. phase transition, glass transition, sintering process etc.) during the thermomechanical analysis, a directly measured “total component” of the length variation or of the coefficient of thermal expansion is additively composed of a “reversible component” and a “non-reversible component”, after determining the “reversible component”, the “non-reversible component” of the length variation or of the coefficient of thermal expansion can be determined simply (by subtracting the reversible component from the total component).
The temperature-dependent determination and characterization of shrinkage or expansion effects of the sample or the influence of such effects on the length variation or the coefficients of thermal expansion is in practice frequently a “main aim” of the TMA.
A generic method for thermomechanical analysis is described, for example, in U.S. Pat. No. 6,007,240 and comprises the following steps:                a) arranging the sample in a thermomechanical analysis device and controlling the temperature of the sample by means of the analysis device according to a modulated temperature program,        b) recording data obtained by means of the analysis device which is representative for a length variation of the sample in the course of the temperature control,        c) evaluating the data in order to determine a reversible component of the length variation and/or a reversible component of the coefficient of thermal expansion of the sample,        d) calculating a corrected reversible component of the length variation by means of a correction parameter which is calculated as the ratio of a total length variation and a reversible component of the length variation.        
FIGS. 1 to 3 show as an example a combination of a linear temperature variation (FIG. 1) and a sinusoidal temperature modulation (FIG. 2) for implementing a modulated temperature program (FIG. 3). In FIGS. 1 to 3 the time t is plotted on the right and the temperature T at the top.
FIGS. 1 to 3 show a linear variation of the temperature T which can be described as follows: T=T0+(β×t). Here T0 denotes an initial temperature and β the heating rate. In the example of FIG. 1 the heating rate β is 2.0 K/min. It should be noted that here and also in the following description of the invention, for simplicity there is always talk of a heating rate β although this heating rate β can also be specified as negative (i.e. corresponding to a “cooling rate”).
FIG. 2 shows a sinusoidal variation of the temperature T which can be described as follows: T=Tavg+AT×sin (ωt). Here Tavg denotes an averaged (over the period duration) temperature. AT denotes an amplitude of the temperature modulation, ω denotes a modulation frequency. In the example of FIG. 2, Tavg is about −1.2° C., AT is 1.5 K and ω is about 2.1 min−1 (corresponding to a period duration of 3.0 min).
FIG. 3 shows the modulated temperature program corresponding to a superposition of the linear variation (FIG. 1) and the sinusoidal variation (FIG. 2), which can be described as follows: T=T0+(β×t)+AT×sin (ωt).
In a TMA thus carried out, all the parameters of the modulated temperature program (here therefore: T0, β, AT and ω) are defined (preset) by a user adapted, for example, to the specific application.
Even in a modulated temperature program, an averaged (over the period duration) temperature Tavg can be specified. For this it holds that: Tavg=T0+(β×t). Thus, the modulated temperature program can be described as follows: T=Tavg+AT×sin (ωt).
If a temperature deviation ΔT=AT×sin (ωt) is defined to describe the modulation-induced “temperature oscillations”, the modulated temperature program can described as follows: T=Tavg+ΔT.
The “coefficient of thermal expansion α” is an important characteristic value which describes the behaviour of a material in relation to variations of its dimensions with temperature variations. The coefficient of thermal expansion α (only dependent on temperature) is a substance-specific material constant which is usually more or less strongly temperature-dependent. The coefficient of thermal expansion α is understood here in the sense of a coefficient of length expansion (unlike the frequently used volumetric thermal expansion coefficient γ) and is the proportionality constant between the temperature variation dT and (reversible) relative length variation dL/L of the material: dL/L=α×dT, Accordingly it holds that:
  α  =            1      L        ×                            d          ⁢                                          ⁢          L                          d          ⁢                                          ⁢          T                    .      
In the situation according to FIG. 1 (linear temperature variation) it holds that dL=L×α×dT and dT=β×dt and therefore: dL=L×α×β×dt.
From this the coefficient of thermal expansion α can be determined as follows:
  α  =            1              L        ⁢                                  ⁢        β              ×                  d        ⁢                                  ⁢        L                    d        ⁢                                  ⁢        T            
The linear profile of the length variation dL/L0 in FIG. 1 as a function of the time t is equivalent to the fact that a in this (idealized) example is a temperature-independent constant.
In the situation according to FIG. 3 (modulated temperature program with linear basic heating), for the length variation there are two components, namely an “underlying component” (caused by the basic heating rate β≠0) and an “oscillating component” (caused by the temperature oscillations). An “underlying dL” can be calculated as an average over the period and is designated hereinafter as dLtotal. Here it holds that dLtotal=Ltotal×α×dTavg and dTavg=β×dt and therefore: dLtotal=Ltotal×α×β×dt.
From this the coefficient of thermal expansion atom, can be determined from dLtotal as follows:
                              α          total                =                              1                                          L                total                            ⁢              β                                ×                                    d              ⁢                                                          ⁢                              L                total                                                    d              ⁢                                                          ⁢              t                                                          (                  Formula          ⁢                                          ⁢          1                )            
The linear behaviour of the length variation Ltotal/L0 as a function of the time t in FIG. 3 is equivalent to the fact that αtotal in this (idealized) example is a temperature-independent constant.
In the situation according to FIG. 2 (temperature modulation) assuming that “T” is the uniform temperature of the relevant material sample, it also holds that ΔL=L×α×ΔT, and ΔT=AT×sin (ωt), from which it follows for the “length variation” ΔL:    ΔL=L×α×AT×sin (ωt).
However, this is only correct for an “ideal” situation in which the sample temperature in each point of the same is the same, i.e. in particular for example in the centre and on the surface of the sample. However, this is not the case in practice since it would assume that the modulation frequency ω is infinitely small or the thermal conductivity of the sample material was infinitely high.
If an amplitude of the length variation is defined as AL to describe the modulation-induced “length oscillations”, it thus holds that:ΔL=AL×sin(αt).
AL is the amplitude of the measured modulated length variations. It can be calculated from a representative measurement signal for these length variations, e.g. by a Fourier analysis (e.g. “Fast Fourier Transformation”).
From this the (reversible) coefficient of thermal expansion αrev can be determined as follows from the oscillating component of the measurement signal theoretically for an ideal situation as follows:
                              α          rev                =                              1            L                    ×                                    A              L                                      A              T                                                          (                  Formula          ⁢                                          ⁢          2                )            
In practice however, the problem manifest in the example of FIG. 2 arises that the temperatures in different areas of the sample differ more or less from one another and from the temperature behaviour which is predefined by a temperature-control device used in the thermomechanical analysis device used according to the modulated temperature program. As a result of the heating and cooling of the sample taking place from outside, in practice a more or less large temperature gradient which cannot be neglected always forms inside the sample.
Thus, in practice, as a result of the not infinitely rapid heat conduction of heat into the sample and out from the sample, this results, for example, in a “lagging” of the same temperature which in FIG. 2 leads to a time shift of the length oscillations (curve dL) relative to the temperature oscillations (curve T). At the same time, this effect has the result that the average value (averaged over entire sample) of the temperature amplitude AT is lower than that predefined by the temperature program. This in turn has the result that the amplitude of the length variation AL is smaller.
In reality the temperature amplitude at the centre of the sample is smaller than the predefined (e.g. by a user by corresponding setting) temperature amplitude AT on the sample surface. Therefore the amplitude of the length oscillations of the sample AL is smaller than in the mentioned ideal situation and the length oscillations are delayed relative to the temperature oscillations. For real modulations a frequency-dependent complex calibration coefficient (correction coefficient) k is required to obtain the corrected reversible coefficient of thermal expansion αrev-corr:αrev-corr=k×αrevΔL=k×Ltotal×αrev×AT×sin(ωt).  (Formula 3)
If a modulated temperature program (e.g. of the type shown in FIG. 3) is used which additionally includes the basic, e.g. linear temperature variation (cf. FIG. 1), such a “calibration” can advantageously be performed so that a calibrated determination of the coefficient of thermal expansion α is possible:
                              α                      rev            -            corr                          =                              1                          L              total                                ×                      1            k                    ×                                    A              L                                      A              T                                                          (                  Formula          ⁢                                          ⁢          4                )            
In general and in particular if the sample is subjected to a thermally induced conversion process at the temperature or temperature variation in the course of the thermomechanical analysis, a (directly measured) “total” length variation dLtotal has a “reversible” component dLrev and a “non-reversible” component dLnonrev, and it holds that: dLtotal=dLrev+dLnonrev.
dLtotal can be calculated as an average (e.g. averaged over precisely one period) of the time-resolved measured value of dL.
dLrev can be calculated as dLrev=dLo rev+Lo×∫ToT αrevdT,    where dLo rev is the reversible length variation at the beginning of the temperature segment. At the beginning of the measurement, it holds that: dLrev=0.
Accordingly a “total” coefficient of thermal expansion αtotal which can be obtained directly from an analysis of dLtotal, has a “reversible” component αrev and a “non-reversible” component αnonrev, and it holds that: αtotal=αrev+αnonrev.
In FIGS. 1 to 3 assuming a purely reversible thermal expansion (dLtotal=dLrev, dLnonrev=0, αtotal=αrev, αnonrev=0), it is therefore equivalent that the sample is not subjected to any thermally induced conversion process in the thermomechanical analysis (e.g. a sample made of metal), in each case the resulting relative length variations dL/L0 are additionally plotted at the top, where L0 denotes an initial length of the sample being analysed (the difference between L and L0 is in practice mostly negligible).
Assuming that in the thermomechanical analysis the sample is not subject to any thermally induced conversion process (αtotal=αrev-corr), the correction parameter k can be calculated from formulas (1) and (3) as the quotient of the “total expansion coefficient” αtotal, which characterizes a “total averaged length variation ΔLavg” (considered over a relatively large temperature variation ΔTavg) and the “reversible expansion coefficient” αrev, which characterizes a “reversible component dLrev of the length variation” dL (considered over a relatively small temperature variation dT):k=αtotal/αrev  (Formula 5)
The quantities αtotal and αrev required to determine the correction parameter k can be obtained as follows:αtotal=1/Ltotal×dLtotal/Tavg and αrev=1/Ltotal×AL/AT 
By means of the correction parameter k determined in this way, a corrected reversible component dLrev-corr of the length variation dL or a corrected reversible component αrev-corr of the (reversible) coefficient of thermal expansion αrev can be calculated (cf. above formula 4).
This method fails however if the sample is subjected to a thermally induced conversion process in the course of the modulated temperature program.